Computer Science > Data Structures and Algorithms

Title:Vertex Sparsifiers and Abstract Rounding Algorithms

Abstract: The notion of vertex sparsification is introduced in \cite{M}, where it was
shown that for any graph $G = (V, E)$ and a subset of $k$ terminals $K \subset
V$, there is a polynomial time algorithm to construct a graph $H = (K, E_H)$ on
just the terminal set so that simultaneously for all cuts $(A, K-A)$, the value
of the minimum cut in $G$ separating $A$ from $K -A$ is approximately the same
as the value of the corresponding cut in $H$.
We give the first super-constant lower bounds for how well a cut-sparsifier
$H$ can simultaneously approximate all minimum cuts in $G$. We prove a lower
bound of $\Omega(\log^{1/4} k)$ -- this is polynomially-related to the known
upper bound of $O(\log k/\log \log k)$. This is an exponential improvement on
the $\Omega(\log \log k)$ bound given in \cite{LM} which in fact was for a
stronger vertex sparsification guarantee, and did not apply to cut sparsifiers.
Despite this negative result, we show that for many natural problems, we do
not need to incur a multiplicative penalty for our reduction. We obtain optimal
$O(\log k)$-competitive Steiner oblivious routing schemes, which generalize the
results in \cite{R}. We also demonstrate that for a wide range of graph packing
problems (which includes maximum concurrent flow, maximum multiflow and
multicast routing, among others, as a special case), the integrality gap of the
linear program is always at most $O(\log k)$ times the integrality gap
restricted to trees. This result helps to explain the ubiquity of the $O(\log
k)$ guarantees for such problems.
Lastly, we use our ideas to give an efficient construction for
vertex-sparsifiers that match the current best existential results -- this was
previously open. Our algorithm makes novel use of Earth-mover constraints.